let $h_n\in C_c^\infty (\mathbb{R}^d)$ s.t. $\int h_n dm = 1$ and $\operatorname{supp}(h_n)\to {0}$.
I've proven that $h_n\to\delta_0$ in $\mathcal{D}'(\mathbb{R}^d)$, now I'm trying to show that for any $f\in L^p(\mathbb{R}^d)$ where $0\le p<\infty$ I get that $f\star h_n\to f$ in $L^p$.
I tried just manually calculating $\|f\star h_n -f\|_p^p$ and showing that it is bound by something which vanishes as $n$ ascends, but not much luck there. I am not even sure if that is the right approach but I don't have any other ideas.
Any hint would be appreciated.
Hint: for $p\in [1,\infty)$
1 - Prove that $C_c(\mathbb{R^d})$ is dense in $L^1(\mathbb{R}^d)$. You can find the proof here, in page 72, Theorem 1.3.20,
2 - By using item 1, prove that $C_c(\mathbb{R}^d)$ is dense in $L^p(\mathbb{R}^d)$,
3 - Prove that if $f\in C(\mathbb{R}^d)$, then $h_n\star f\rightarrow f$ uniformly on compact sets of $\mathbb{R}^d$,
4 - By using item 2 and 3, you can conclude.